Everything you want is along the Ring road. It has different names for different sections - Universitaetsring, Burgring, Opernring, etc.

Essentially just a ring enclosing the inner district. Along the ring are all the opera houses, theater houses, concert hall, parliament, university, museums, palaces, city hall, beautiful parks with monuments of mozart schubert etc.

Famous coffee house such as Landtmann is also there. You can burn a lot of time just walking along it, venturing into the (large) park/garden of your choice, admire the buildings from the outside, or even wander inside parts of the Neo Gothic city hall. In spring and summer chances are there is some festival/event outside it as well.

"Lounge" in the public space around the museum quarter. Eat or drink in all the places along there. Take a nap under the sun in the park such as Sigmund Freud park; it has public lounge chairs with lots of students sunbathing and relaxing during the day.

Head back to airport around sunset. Thats a good day